# Hubbard–Stratonovich transformation

The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its respective field theory by linearizing the density operator in the many-body interaction term of the Hamiltonian and introducing a scalar auxiliary field. It is defined via the integral identity[1] [2]

$\exp \left\{ - \frac{a}{2} x^2 \right\} = \sqrt{\frac{1}{2 \pi a}} \; \int_{-\infty}^\infty \exp \left[ - \frac{y^2}{2 a} - i x y \right] \, dy,$

where the real constant $a > 0$. The basic idea of the HS transformation is to reformulate a system of particles interacting through two-body potentials into a system of independent particles interacting with a fluctuating field. The procedure is widely used in polymer physics, classical particle physics, spin glass theory, and electronic structure theory.

## Calculation of resulting field theories

The resulting field theories are well-suited for the application of effective approximation techniques, like the mean field approximation. A major difficulty arising in the simulation with such field theories is their highly oscillatory nature in case of strong interactions, which leads to the well-known numerical sign problem. The problem originates from the repulsive part of the interaction potential, which implicates the introduction of the complex factor via the HS transformation.

## References

1. ^ Стратонович, Р. Л. (1957). Об одном методе вычисления квантовых функций распределения. Доклады АН СССР (in Russian) 115 (6): 1097–1100. Translation available: Stratonovich, R.L. (1958). "On a Method of Calculating Quantum Distribution Functions". Soviet Physics Doklady 2: 416. Bibcode:1957SPhD....2..416S.
2. ^ Hubbard, J. (1959). "Calculation of Partition Functions". Physical Review Letters 3 (2): 77. Bibcode:1959PhRvL...3...77H. doi:10.1103/PhysRevLett.3.77.